LieAlgebras[Codifferential] - calculate the codifferential of a multi-vector defined on a Lie algebra with coefficients in a representation
Calling Sequences
Codifferential(Z)
Parameters
Z - a multi-vector defined on a Lie algebra, or on a Lie algebra with coefficients in a representation V
Description
Examples
Let 𝔤 be a Lie algebra. The codifferential ∂ of monomial bi-vectors and tri-vectors on 𝔤 is defined by
∂ x1 ∧ x2 = x1, x2 and ∂ x1 ∧x2 ∧ x3 = x1, x2 ∧x3 − x1 ,x2 ∧x3 + x2, x3 ∧x1 .
The formula for a general monomial multi-vector is
∂ x1 ∧ x2 ∧⋅⋅⋅ xp =∑i<j−1i +j +1xi , xj∧ x1 ∧ ⋅⋅⋅ ∧ xi‾ ∧ ⋅⋅⋅ ∧xj‾ ∧⋅⋅⋅ xp
where the barred vectors are omitted from the wedge product. A general multi-vector of degree p is a superposition of monomials of degree p. The definition of the codifferential is extended to all multi-vectors by linearity.
Let ρ: 𝔤 →glV be a representation of 𝔤 on a vector space V. For x ∈ 𝔤 and w ∈ V, write ρxw = x ⋅w. For multi-vectors with coefficients in V, the above formulas for the codifferential are amended to
∂w x1 ∧ x2 = x1⋅wx2 − x2 ⋅wx1 + wx1, x2,
∂w x1 ∧x2 ∧ x3 =x1`⋅wx2∧x3 − x2 ⋅wx1∧x3 + x3⋅wx1∧x2 + x1, x2 ∧x3 − x1 ,x2 ∧x3 + x2, x3 ∧x1 and, in general,
∂w x1 ∧ x2 ∧⋅⋅⋅ xp =∑i=1p−1i +1xi ⋅w∧ x1 ∧ ⋅⋅⋅ ∧ xi‾ ∧⋅⋅⋅ xp + ∑i<j−1i +j +1xi , xj∧ x1 ∧ ⋅⋅⋅ ∧ xi‾ ∧ ⋅⋅⋅ ∧xj‾ ∧⋅⋅⋅ xp.
Again, these definitions are extended to all multi-vectors by linearity.
The command Codifferential computes the codifferential of a multi-vector Z. Note that if Z has degree p, then ∂Z has degree p−1.
The co-differential satisfies ∂2 =0 It commutes with the Lie derivative Z and satisfies, for any vector X, 𝒵XZ = X ∧∂Z + ∂ X ∧Z .
withDifferentialGeometry:withLieAlgebras:
Example 1.
First initialize a 5-dimensional Lie algebra.
LD1≔LieAlgebraDatax2,x3=x1,x2,x5=x3,x4,x5=x4,x1,x2,x3,x4,x5,alg
LD1:=e2,e3=e1,e2,e5=e3,e4,e5=e4
DGsetupLD1
Lie algebra: alg
Define a bi-vector and calculate its codifferential.
Z≔evalDGae2&we3+be2&we5+ce2&we4
Z:=ae2⋀e3+ce2⋀e4+be2⋀e5
CodifferentialZ
ae1+be3
Define a tri-vector and calculate its codifferential.
Z≔evalDGae2&we3&we4+be3&we4&we5
Z:=ae2⋀e3⋀e4+be3⋀e4⋀e5
W≔CodifferentialZ
W:=ae1⋀e4−be3⋀e4
Check that ∂2Z =0.
CodifferentialW
0e1
Example 2.
In this example we calculate the codifferentials for some multi-vectors defined on a Lie algebra with coefficients in a representation. For this example we shall use the Lie algebra so4and its standard 4-dimensional representation. To create the computational environment we use the commands SimpleLieAlgebraData, StandardRepresentation and Representation.
LD2≔SimpleLieAlgebraDataso(4),so4
LD2:=e1,e2=e4,e1,e3=e5,e1,e4=−e2,e1,e5=−e3,e2,e3=e6,e2,e4=e1,e2,e6=−e3,e3,e5=e1,e3,e6=e2,e4,e5=e6,e4,e6=−e5,e5,e6=e4
DGsetupLD2
Lie algebra: so4
A≔StandardRepresentationso4
Create a 4-dimensional vector space to serve as the representation space.
DGsetupw1,w2,w3,w4,V
frame name: V
ρ≔Representationso4,V,A
Initialize the Lie algebra so4 with coefficients in the standard representation.
DGsetupso4,ρ,so4V
Lie algebra with coefficients: so4V
Calculate the codifferential of a bi-vector.
Z≔evalDGw1e1&we2
Z:=w1e1⋀e2
−w3e1+w2e2+w1e4
Calculate the codifferential of a multi-vector of degree 4.
Z≔evalDGw4e1&we2&we5&we6
Z:=w4e1⋀e2⋀e5⋀e6
W:=w4e1⋀e2⋀e4+w3e1⋀e2⋀e5−w2e1⋀e2⋀e6−w4e1⋀e3⋀e5−w4e2⋀e3⋀e6+w4e4⋀e5⋀e6
0e1⋀e2
See Also
DifferentialGeometry
LieAlgebras
Adjoint
ExteriorDerivative
Representation
SimpleLieAlgebraData
StandardRepresentation
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